metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.48D28, (C2×C8)⋊21D14, (C2×D28)⋊14C4, (C2×C56)⋊36C22, D28.24(C2×C4), (C2×C4).153D28, (C2×C28).173D4, C28.417(C2×D4), C2.D56⋊39C2, C4.13(D14⋊C4), C2.4(C8⋊D14), (C2×M4(2))⋊11D7, C4⋊Dic7⋊48C22, C22.57(C2×D28), C28.27(C22⋊C4), C14.20(C8⋊C22), (C14×M4(2))⋊19C2, (C2×C28).773C23, C28.115(C22×C4), (C22×D28).15C2, (C22×C4).139D14, (C22×C14).101D4, C7⋊3(C23.37D4), C22.28(D14⋊C4), (C2×D28).200C22, C23.21D14⋊16C2, (C22×C28).188C22, C4.73(C2×C4×D7), (C2×C4).53(C4×D7), C2.30(C2×D14⋊C4), C4.110(C2×C7⋊D4), (C2×C28).108(C2×C4), (C2×C14).163(C2×D4), (C2×C4).76(C7⋊D4), C14.58(C2×C22⋊C4), (C2×C4).722(C22×D7), (C2×C14).21(C22⋊C4), SmallGroup(448,665)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.48D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd27 >
Subgroups: 1316 in 190 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C56, D28, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×C14, C23.37D4, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C2×D28, C2×D28, C22×C28, C23×D7, C2.D56, C23.21D14, C14×M4(2), C22×D28, C23.48D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C8⋊C22, C4×D7, D28, C7⋊D4, C22×D7, C23.37D4, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C8⋊D14, C2×D14⋊C4, C23.48D28
►On 112 points
Generators in S112
(2 30)(4 32)(6 34)(8 36)(10 38)(12 40)(14 42)(16 44)(18 46)(20 48)(22 50)(24 52)(26 54)(28 56)(58 86)(60 88)(62 90)(64 92)(66 94)(68 96)(70 98)(72 100)(74 102)(76 104)(78 106)(80 108)(82 110)(84 112)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 58 59 56)(2 55 60 57)(3 112 61 54)(4 53 62 111)(5 110 63 52)(6 51 64 109)(7 108 65 50)(8 49 66 107)(9 106 67 48)(10 47 68 105)(11 104 69 46)(12 45 70 103)(13 102 71 44)(14 43 72 101)(15 100 73 42)(16 41 74 99)(17 98 75 40)(18 39 76 97)(19 96 77 38)(20 37 78 95)(21 94 79 36)(22 35 80 93)(23 92 81 34)(24 33 82 91)(25 90 83 32)(26 31 84 89)(27 88 85 30)(28 29 86 87)
G:=sub<Sym(112)| (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,58,59,56)(2,55,60,57)(3,112,61,54)(4,53,62,111)(5,110,63,52)(6,51,64,109)(7,108,65,50)(8,49,66,107)(9,106,67,48)(10,47,68,105)(11,104,69,46)(12,45,70,103)(13,102,71,44)(14,43,72,101)(15,100,73,42)(16,41,74,99)(17,98,75,40)(18,39,76,97)(19,96,77,38)(20,37,78,95)(21,94,79,36)(22,35,80,93)(23,92,81,34)(24,33,82,91)(25,90,83,32)(26,31,84,89)(27,88,85,30)(28,29,86,87)>;
G:=Group( (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,58,59,56)(2,55,60,57)(3,112,61,54)(4,53,62,111)(5,110,63,52)(6,51,64,109)(7,108,65,50)(8,49,66,107)(9,106,67,48)(10,47,68,105)(11,104,69,46)(12,45,70,103)(13,102,71,44)(14,43,72,101)(15,100,73,42)(16,41,74,99)(17,98,75,40)(18,39,76,97)(19,96,77,38)(20,37,78,95)(21,94,79,36)(22,35,80,93)(23,92,81,34)(24,33,82,91)(25,90,83,32)(26,31,84,89)(27,88,85,30)(28,29,86,87) );
G=PermutationGroup([[(2,30),(4,32),(6,34),(8,36),(10,38),(12,40),(14,42),(16,44),(18,46),(20,48),(22,50),(24,52),(26,54),(28,56),(58,86),(60,88),(62,90),(64,92),(66,94),(68,96),(70,98),(72,100),(74,102),(76,104),(78,106),(80,108),(82,110),(84,112)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,58,59,56),(2,55,60,57),(3,112,61,54),(4,53,62,111),(5,110,63,52),(6,51,64,109),(7,108,65,50),(8,49,66,107),(9,106,67,48),(10,47,68,105),(11,104,69,46),(12,45,70,103),(13,102,71,44),(14,43,72,101),(15,100,73,42),(16,41,74,99),(17,98,75,40),(18,39,76,97),(19,96,77,38),(20,37,78,95),(21,94,79,36),(22,35,80,93),(23,92,81,34),(24,33,82,91),(25,90,83,32),(26,31,84,89),(27,88,85,30),(28,29,86,87)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | D28 | C8⋊C22 | C8⋊D14 |
| kernel | C23.48D28 | C2.D56 | C23.21D14 | C14×M4(2) | C22×D28 | C2×D28 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 6 | 3 | 12 | 6 | 12 | 6 | 2 | 12 |
Matrix representation of C23.48D28 ►in GL8(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 112 |
| 13 | 77 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 94 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 103 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 91 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 112 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 112 | 0 |
| 67 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 80 | 46 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 91 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 112 | 112 |
| 0 | 0 | 0 | 0 | 72 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 112 | 0 |
G:=sub<GL(8,GF(113))| [112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[13,32,0,0,0,0,0,0,77,94,0,0,0,0,0,0,0,0,10,13,0,0,0,0,0,0,88,103,0,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,91,112,112,112,0,0,0,0,0,1,0,0],[67,80,0,0,0,0,0,0,23,46,0,0,0,0,0,0,0,0,103,5,0,0,0,0,0,0,25,10,0,0,0,0,0,0,0,0,1,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,91,112,112,112,0,0,0,0,0,112,0,0] >;
C23.48D28 in GAP, Magma, Sage, TeX
C_2^3._{48}D_{28} % in TeX
G:=Group("C2^3.48D28"); // GroupNames label
G:=SmallGroup(448,665);
// by ID
G=gap.SmallGroup(448,665);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,387,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^27>;
// generators/relations